Genus 2 curve 188356.a.376712.1

• Label: 188356.a.376712.1
• Conductor: $188356$
• Discriminant: $376712$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x + 1)y = x^5 - 2x^4 - 6x^3 - 3x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(188356\)	\(=\)	\( 2^{2} \cdot 7^{2} \cdot 31^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(376712\)	\(=\)	\( 2^{3} \cdot 7^{2} \cdot 31^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(964\)	\(=\)	\( 2^{2} \cdot 241 \)
\( I_4 \)	\(=\)	\(26257\)	\(=\)	\( 7 \cdot 11^{2} \cdot 31 \)
\( I_6 \)	\(=\)	\(7791385\)	\(=\)	\( 5 \cdot 7 \cdot 31 \cdot 43 \cdot 167 \)
\( I_{10} \)	\(=\)	\(48219136\)	\(=\)	\( 2^{10} \cdot 7^{2} \cdot 31^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((3 : -4 : 1)\)	\((-1 : -12 : 4)\)	\((-4 : 25 : 3)\)	\((3 : -27 : 1)\)	\((-1 : -35 : 4)\)	\((-4 : 48 : 3)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.208232\)	\(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.208232\)	\(\infty\)

2-torsion field: 6.6.24109568.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.032520 \)
Real period:	\( 14.03010 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.368801 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(3\)	\(3\)	\(1 + T + T^{2}\)
\(7\)	\(2\)	\(2\)	\(1\)	\(1 + 5 T + 7 T^{2}\)
\(31\)	\(2\)	\(2\)	\(1\)	\(1 + 4 T + 31 T^{2}\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.40.3	no
\(3\)	3.480.12	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_6$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.481170140857.1 with defining polynomial:
  \(x^{6} - x^{5} - 90 x^{4} + 237 x^{3} + 1408 x^{2} - 2640 x - 7811\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{31616709}{2224} b^{5} - \frac{508297487}{8896} b^{4} - \frac{9846895747}{8896} b^{3} + \frac{59702831507}{8896} b^{2} - \frac{1112605409}{4448} b - \frac{327094910479}{8896}\)
  \(g_6 = -\frac{6809701164035}{71168} b^{5} + \frac{13686477866341}{35584} b^{4} + \frac{8284610658751}{1112} b^{3} - \frac{3214989603474019}{71168} b^{2} + \frac{120237864274395}{71168} b + \frac{17614540690886321}{71168}\)
   Conductor norm: 64

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.481170140857.1 with defining polynomial \(x^{6} - x^{5} - 90 x^{4} + 237 x^{3} + 1408 x^{2} - 2640 x - 7811\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{217}) \) with generator \(-\frac{3}{139} a^{5} - \frac{16}{139} a^{4} + \frac{215}{139} a^{3} + \frac{697}{139} a^{2} - \frac{3192}{139} a - \frac{8404}{139}\) with minimal polynomial \(x^{2} - x - 54\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.47089.1 with generator \(\frac{9}{2224} a^{5} + \frac{3}{139} a^{4} - \frac{253}{1112} a^{3} - \frac{1813}{2224} a^{2} + \frac{1931}{2224} a + \frac{29243}{2224}\) with minimal polynomial \(x^{3} - x^{2} - 72 x - 209\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple